Modeling Pigmented Materials for Realistic
Image Synthesis
CHET
S. HAASE
University
and
GARY
W. MEYER
of Oregon
This article discusses and applies the Kuhelka Munk theory of pigment mixing to computer
graphics in order to facilitate improved image synthesis. The theories of additive and subtractive
color m]xing arc discussed and are shown to be insufficient for pigmented materials. The
Kubclka Munk theory of pigment mixing is developed and the relevant equations arc derived.
pigmer]t mixing experiments are performed and the results are displayed on cokw tekm’ision
monitors. A paint program that uscs Kubelka Munk theory to mix real pigments is presented.
Theories of color matching with pigments are extended to determine rcflectances for use in
realistic image synthesis.
(’categories and Subject Descriptors: 1.3.4 [Computer
.s,v,<tcm.s:1.17 [Computer Graphics]: Three-Dimensional
Graphics]: Graphics
Graphics
Utilities
and Realism
color,
patnt
.shad-
jf),qr,
.shfldof
IIIIg,
and text
MI.(,
(;cneral
Terms: Al~oritbms,
Experimentation,
Additi[,nal Key Words and Phrases:
modeling. pigment mixing
Human Factors
Color matching,
color science, cnlor selection,
illumination
1, INTRODUCTION
The illumination model in computer graphics is constantly being refined so
that it provides an ever more accurate model of reality. However, no illumination model will work unless the objects in the scene have been properly
described. This fact has been realized and the surface properties of objects
have been painstakingly
modeled, especially as they relate to reflection
direction and strength. But a comprehensive
approach to object modeling
should also take into account the physical makeup of those surfaces. In
particular, most manufactured
objects in the real world have pigmented
surfaces, such as painted items, plastic objects, and textiles. To accurate] y
This research was funded by a grant from Apple Computer, Inc. Additional support was provided
by the Hewlett-Packard
(hmpany and the Oregon Advanced Computing Institute,.
.kuthors’ address: Department of Computer and Information Science, University of Oregon,
Eugene, OR 97403.
Permission to copy without fec all or part of this material is granted provided that the copies are
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of tbe publication and its date appear, and notice is given that copying is by permission of the
Association for Computing Machinery. To copy otherwise, or to republish, requires a fee and/or
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c 1992 ACM 0730-0301/92/
1000-305$01.50
ACM Transactions on Graphics, Vol. 11. No. 4, octoher 1992, Pages 305 335
306
.
C. S. Haase and G. W. Meyer
model the color of any pigmented object, the physical interactions within the
pigmented surface must be included.
A straightforward
way of specifying color in a graphics scene is via the
common red, green, and blue ( RGB ) triplet. This approach is the easiest to
deal with in scene generation computations, since this specification is exactly
what is needed by the computer to generate colors for the screen output. But
this method is far removed from reality—the
RGB method is only appropriate for additive colorants, such as colored lights (e.g., the phosphors in the
monitor screen); nonemitting
objects should be specified in terms of their
reflective and transmissive characteristics.
A second approach for specifying color is in cyan, magenta, and yellow
( CiWY ) space. This specification is in the domain of subtractive color synthesis and accurately models the effect of light transmission through a colored
surface. This specification is especially important with respect to the color
reproduction industry. For instance, color printer manufacturers
must deal
with translating images from the domain of computer monitors (in which all
colors are RGB ) to the domain of hardcopy devices, in which the colors must
be specified in CMY coordinates (Stone, et al., 1988). This subtractive method
is also used in some screen-based painting applications, such as Small (1991)
and some commercially
available programs, as a better attempt to model
realistic color mixing. However, as will be shown in this paper, this method is
insufficient for representing pigmented surfaces, since CMY color synthesis
works best for purely transmitting materials; pigmented surfaces have both
transmitting and reflecting characteristics.
The most accurate way of dealing with light in a synthetic imaging
application is on a wavelength-by-wavelength
basis [8, 14, 19]. It is only in
this way that the more subtle illumination
modeling and color calculation
problems can be handled. The spectral energy distributions
of the light
sources in the environment
must be given and the spectral reflectance,
transmittance, and absorptance of the surfaces with which these light sources
interact must be specified. Pigment mixing is an example of an optical
phenomenon which must be modeled in this way. The Kubelka-Munk
theory
is used by the paint industry to predict the reflectance that will result when
two or more pigments are mixed. This theory has been discussed within a
computer graphics context [12] but has not been used to solve color synthesis
problems.
In this paper we introduce the Kubelka-Munk
theory of pigment mixing
and apply that theory to computer graphics to facilitate correct-color calculation for realistic image generation. Section 2 develops some of the significant
differences in additive and subtractive color mixing and discusses the need
for different mixing theory for pigmented materials. Section 3 introduces and
develops the Kubelka-Munk
theory of pigment mixing. Section 4 discusses
the pigment-mixing
experiments carried out for this project. Section 5 introduces the topic of color matching with respect to pigments; given a sample
color, it is possible to calculate a pigment mixture that will render an
approximation to this color. The final section provides a summary.
ACM Transactions on Graphics, Vol. 11, No. 4, October 1992
Fragmented Materials for Realistic Image Synthesis
Fig. 1.
307
Color television monitor gamut in CIE XYZ space
2. SUBTRACTIVE COLOR MIXING
The human visual system contains three types of daylight receptors, called
cones. The spectral sensitivity functions of the three cone types are different,
but they do overlap significantly. Because there are only three types of cones
to sample the entire visible spectrum, it is possible for two different spectral
energy distributions to produce the same three cone signals. When this
happens the visual system is unable to distinguish between the color sensations evoked, and the spectral energy distributions are said to be metamers.
This means that satisfactory color reproduction can be achieved without
having to perfectly reconstruct the original spectral energy distribution. In
fact, the number, shape, and position of the spectral sensitivity functions
make it possible to reproduce most colors by manipulating only the short,
medium, and long wavelength regions of the spectral energy distributions
used to create the reproduction.
The most straightforward way of controlling the energy in these three
different portions of the spectrum is to combine the light from three sources.
This approach, which is called additive color synthesis, is used in the standard cathode ray tube of color televisions and monitors where three electron
guns independently address the short, medium, and long (blue, green, and
red) areas of the spectrum. For instance, in combining a light from the short
wavelength region with one from the medium wavelength region, a spectral
energy distribution is produced which elicits the visual sensation of cyan. The
color television monitor gamut produced by additively mixing the light from
the three red, green, and blue primaries is shown in Figure 1.
A more indirect way of controlling the energy in the three different spectral
regions is to place a stack of three filters in front of a light source. This
approach is called subtractive color synthesis, because information is removed from the spectral energy distribution of the light source. This acACM Transactions on Graphics, Vol. 11. No. 4. Octohcr 1992.
308
.
C, S. Haase
and G. W. Meyer
complishes the same goal as additive color synthesis but does not allow
completely independent control of the short, medium, and long wavelength
regions of the spectrum. For instance, suppose two subtractive colorants are
used—cyan and yellow—the
cyan colorant transmits energy in the medium
and short regions of the spectrum (similar to the cyan produced in the
additive example above), and the yellow passes energy in the long and
medium regions. By combining these colorants in subtractive color mixing
(further explained below), we derive a colorant which controls energy in the
medium region and we see a green color; essentially, the short and long
spectral energy present in both components was subtracted from the final
result. Again, the visual system is being stimulated by selectively manipulating particular regions of the visible spectrum.
The example of the colored filter given above is a very simplified view of
subtractive color. Instead of talking about the long, medium, and short areas
of the visible spectrum, we can quantify the characteristics
of a given
subtractive colorant on a wavelength-by-wavelength
basis. In particular, at
each wavelength of the visible spectrum, the colorant will allow some given
percentage of light energy to pass through. This information, plotted graphically with wavelength versus the percentage of light passed through the
material, gives us the transmittance curve for that filter. An example of such
a curve is seen in Figure 2, where we see that more light energy toward the
lower end of the spectrum is allowed to pass than light energy at the higher
end. Thus, given a light source which emits equal energy at all wavelengths,
the resulting color of the light passing through the example filter would be
cyan.
These subtractive colorants are made from some clear material in which a
colored substance is dissolved at a given concentration.
The transmittance
data discussed above assumes a given thickness of the colorant layer, h, and
a given concentration, c, of the colorant. If we change either of these variables
—the thickness or the colorant concentration—we
would expect the nature of
the transmittance of the filter to change as well. These transmittance changes
are described by the Bouguer and Beer laws. Bouguer’s law states that given
a transmittance
T at a particular wavelength and a thickness h, the transmittance at a thickness of nh at that wavelength is Tn. Similarly, Beer’s law
states that given a transmittance
T at a particular colorant concentration
c
and a constant thickness, the transmittance at that wavelength is Tm where
the concentration
is mc [10]. The effect of varying the concentration
of a
hypothetical dye can be seen in Figure 2.
Changing the concentration or the thickness of a filter produces a locus of
tristimulus values which is much different than the collection of tristimulus
values created by varying the intensity of a light source that shines through a
filter with constant transmittance
characteristics.
Given a filter that corresponds to unit concentration of the dye shown in Figure 2, Figure 3 depicts
the locus of CIE XYZ tristimulus values produced by changing the intensity
of the light passing through the filter. This locus is a straight line that
projects to a single point on the unit plane. This means that a single pair of
chromaticity coordinates can be used to describe the colors that are produced.
ACM Transactions on Graphics, Vol. 11, No. 4, October 1992
Fragmented Materials for Realistic Image Synthesis
Varying
I
I
T
Concentration
Cyan
.
309
Cone Chart
= 1
1
I
1
.
..–—
400
—— .
450
500
550
600
650
700
Wavelength
Fig. 2.
Spectral transmittances
for unit thickness
and varying concentration
of a cyan dye.
On the other hand, when the light source remains constant and the concentration of the filter varies, the locus of CIE XYZ tristimulus values shown in
Figure 4 results. The projection of this space curve onto the unit plane
produces a curved line segment. This indicates that the color produced by the
filter is changing and no single pair of chromaticity
coordinates can fully
describe the situation.
Calculation of subtractive colorant mixtures is analogous to subtractive
concentration calculations, as above. Whereas, in the case of varying concentrations, we multiply the transmittance
at every wavelength by itself for
every unit of concentration, subtractive mixtures require the multiplication of
transmittances of the different colorants in the mixture. It is helpful to think
of this process in terms of separate colored filters. Light passing through each
filter is multiplied by the transmittance
of that filter (taking into account
whatever concentration
of thickness that filter has), then the process is
repeated for each additional filter (and color). In Figure 5 we see the effect of
mixing the cyan dye from Figure 4 with two other dyes. The gamut produced
by varying the individual concentrations
of the colorants is substantially
different from the additive color-television
monitor gamut shown in Figure 1.
The difference in shape between the additive and subtractive gamuts illustrates the linear versus nonlinear dichotomy that exists between these two
approaches to color mixing.
Water color paints can be modeled using subtractive color mixing, because
the paint is usually transparent enough to allow light to pass to the substrate
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C. S. Haase and G. W. Meyer
Fig. 3. Locus of values in CIE
XYZ space resulting from shining
a light of varying intensity
through a cyan filter with constant concentration. The intersection with the unit plane is a
single point.
Fig. 4. Locus of values in CIE XYZ space resulting from
shining a light of constant intensity through a cyan filter
of varying concentration. The intersection with the unit
plane is a curved line.
Fig. 5. Gamut produced in CIE XYZ space by mixing
varying concentrations of cyan, yellow, and magenta dyes.
and reflect back through the paint appropriately colored with respect to the
transmitting properties of the watercolor [lo]. Pigmented materials such as
artists’ oil paints, on the other hand, do not work as well within the
subtractive domain. The preceding discussion assumed that the materials
involved were transparent and allowed light to pass through the substance,
ACM Transactions on Graphics, Vol. 11, No. 4. October 1992
Fragmented Materials for Realistic Image Synthesis
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311
I
I
Fig. 6.
Light interactions
within a pigmented
material (after Evans, 1948),
selectively absorbing light energy at certain wavelengths;
but pigmented
solutions consist of opaque particles in a transparent medium, quite different
from the completely transparent solutions of subtractive colorants.
To understand pigmented solutions, we must look at them on a particle
level. Each pigment particle has both absorbing and reflecting properties.
The absorption of a pigment particle is related to its transmittance, because
absorption is equal to 1 minus the transmittance.
But in addition to this
absorption effect, there is the effect of selective reflectance, or scattering, of
light energy. Scattering is the property that determines how much of the
incident light is reflected from the surface of the pigment particle per
wavelength. Each of these two factors (illustrated in Figure 6) can be described by spectral curves which describe the percentage of energy absorbed
or scattered by the pigmented solution at each visible wavelength. Together
these absorption and scattering properties produce spectral reflectance characteristics which create the color that we observe.
The problem in making calculations based on these properties is that each
pigment particle in the solution has the properties of scattering and absorption. In the case of the transparent
solutions discussed previously,
the
colorant was completely dissolved in the solution or material. But in the case
of pigments, the pigment material is ground up finely and distributed
throughout some filler, such as linseed oil in the case of artists’ oil paints.
Making calculations based on the independent properties of scattering and
absorption for every pigment in the solution would require accounting for the
size, density, and shape of every particle. One of the theories of scattering
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C. S. Haase and G. W. Meyer
and absorption of materials, the Mie theory, uses this information to calculate various properties of individual pigment particles. But this level of detail
is somewhat too complex for calculating information about pigment solutions
[17]. Some other means to calculate the absorption and scattering for the
overall material would be very beneficial. This, then, was the motivation for
the work done by Kubelka and Munk.
3. KUBELK+MUNK
THEORY
Instead of looking at pigmented solutions on a particle level, Kubelka and
Munk observed the effects of light energy in the entire solution. This approach incorporated
the ideas behind the pigment particle scattering and
absorption interactions, but allowed a much easier and more comprehensive
calculation of entire pigmented systems. By using this approach, Kubelka
and Munk came up with the equations that are still used in the pigment
industry today.
Kubelka and Munk examined what happens as light traverses a thin layer
of paint applied over a substrate. At any location in the paint, light from the
surface is moving deeper into the material and light that has reflected from
the substrate is traveling back toward the top of the film. A certain fraction,
K, of the light traveling
in each direction will be absorbed by the material,
and another portion, S, will be scattered. Light from each direction that is
scattered is assumed to contribute to the amount of light traveling
in the
opposite direction. A set of differential equations were written as a result of
this analysis. A complete derivation of these equations is given in the
Appendix.
For complete hiding (i.e., when the pigment layer is so thick that the
substrate cannot be seen), the solution of the differential equations for the
reflectance, R=, is
/( )
K2
R.=l+;
This is often rewritten
–
F
K
+2F.
(1)
as:
K
()z=
(1 -
R.)2
2R.
“
(2)
Using Equation 1, we can now derive the spectral reflectance of any pigmented material at complete hiding if we know its respective K and S values
(and the spectral properties of the solution in which the pigments are
immersed). These are the equations that were used for all of the work
reported in this paper. Solutions to the differential equations can also be
found for the case in which the spectral characteristics
of the base material
affect the spectral composition of the entire system [ 16].
ACM Transactions on Graphics, Vol. 11, No. 4, October 1992
Fragmented Materials for Realistic Image Synthesis
We can find the properties of mixtures of pigmented solutions by using
fact that combinations of absorption and scattering are linear [9]:
K,v =
313
the
~ K, c,
(3)
Sw= ~ s,c,
,=1
(4)
1–1
and
K
(-)s
X:
,K, c,
~= L; ,S,c,
(5)
where
Kiv
absorption
of pigment
SM
scattering
of pigment
n
number of pigments
(’,
concentration
K,
absorption
of i th pigment
s,
scattering
of ith pigment.
mixture
mixture
in mixture
of i th pigment in mixture by weight of dry pigment
Each equation is calculated for each wavelength. The use of these equations
in predicting mixtures and matches of colors is covered in later sections of
this paper.
It is helpful at this point to look at the K and S values more closely in
terms of their significance and their calculation. First of all, the units of K
and S are not important for the purposes of these equations. As they are
always used in conjunction with each other for the Kubelka–Munk
equations,
their significance
is in their ratio. This simplifies the gathering of the
absorption and scattering data significantly; instead of having to measure the
actual units of K and S for any particular sample, we can use the equations
above and spectral reflectance curves to derive unitless K and S values [ 15].
For example, to derive the K and S values for the titanium dioxide
pigment used throughout this work, S was set equal to 1 at all wavelengths
and K was determined by using a given spectral reflectance curve for R [ 11]
and Equation 2 above [ 15]. Once the K and S data for one pigment (titanium
dioxide, in this case) has been calculated, the K and S for any given pigment
may be derived, using the equations above and data for the spectral reflectance
of various mixtures of the known pigment with the pigment in
question. In the case of cadmium yellow medium, one of the primary pigments used in this project, two spectral curves were used [ 11]: one of
cadmium yellow medium as a masstorze (i.e., a pure pigment) and one of
cadmium yellow medium as a tint (i.e., mixed with titanium dioxide in a
given ratio). By using Equations 2 and 5 above, and knowing the K and S
values for titanium dioxide and the relative concentrations of the pigments in
A(7.M Transact][ms IInGraphics, Vol. 11, N(I. 4, octoh~,r 19%?
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C. S. Haase and G. W. Meyer
the tinted sample, there were two equations in two unknowns, and the K and
S values for cadmium yellow medium were solved for appropriately.
Again, the actual units of K and S are irrelevant as long as the relative
proportions of the factors are correct. However, pigment data gathered from
different sources must be used with caution. For instance, if the data for two
different pigments were gathered from different sources (perhaps computed
using two different sets of K and S data for titanium dioxide), the pigments
should not be used together in the same equations. Unfortunately,
the data
required by the Kubelka–Munk
equations must be extremely accurate, and
gathering such data can be very difficult. The method of calculating K and S
given above is very straightforward,
but those K and S values will only be as
good as the spectral curves on which they were based. Moreover, the data
must be consistent across pigments. For example, if titanium dioxide is used
as the tinting agent in the K and S calculations, any insufficiencies
in the
data for titanium dioxide will be reflected in unsatisfactory data for all other
pigments that used these values in their own K and S calculations.
There are many areas in the process of data gathering which must be
handled carefully. To begin with, the measurement of pigment data must be
consistent with its proposed use. For instance, since this research uses oil
paints for examples, we gathered data from oil paints. Spectral curves that
describe the reflectance
for those pigments in other solutions are insufficient
for our purposes. Also, the mixing of the pigment compounds is important;
the ratio of pigment to solution as well as the ratio of pigments in tints could
affect the outcome. Application of the pigment compounds to the measurement background is also problematic; a layer of the pigmented solution which
does not completely hide the background will be affected by that background
color. Measuring the spectral curve for the mixture is another important
area. The accuracy of the measuring device in addition to the suitability of
the lighting and viewing conditions can strongly affect the outcome of the
measurements.
Several simplifying assumptions were made in the theory and equations
developed by Kubelka and Munk. First of all, the pigmented solution is
treated as a uniform material, assuming complete dispersion of pigment in
the solution and homogeneous density of pigment particles. In reality, various forms of deviations from this idea can occur, including flocculation
(clumping) and floating of the pigment particles [211. Second, diffuse lighting
and viewing conditions are assumed, and there is no account for surface
reflection. Finally, the pigment surface is assumed to be planar, with no light
escaping from the edges of the surface. This case is seen in the above
assumption that light energy is either undisturbed, absorbed, or scattered.
These assumptions are discussed more fully in [4] and [21].
Further work has been done on the Kubelka–Munk
equations and its
shortcomings.
An important example is the work of Saunderson
[24], in
which a formula was proposed which attempts to account for both external
and internal surface reflection. This formula is used to adjust the measured
reflectance
from which K and S are determined. Given the Fresnel reACM Transactions on Graphics, Vol.
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4, October 1992
Fragmented Materials for Realistic Image Synthesis
.
315
tlectance equation:
(6)
where
n,
refraction
index of the external
medium (i.e., the air)
nz
refraction
index of the internal
medium (i.e., the oil).
Saunderson’s
formula is as follows:
R“, –kl
R,=kl+
1 –kl
–ky
+kzR,~
(7)
where
R,
theoretical reflectance;
mining K and S
R,,,
measured
tometer
reflectance;
k[
front-surface
kz
internal reflectance
4. PIGMENT
spectral
reflectance
spectral reflectance
reflectance
adjusted
measured
for use in deterby a spectropho-
of the film
of the film.
MIXING
Given a set of pigments with absorption and scattering data, we can calculate
the absorption and scattering curves for any specified mixture of the pigments using Equations 3 through 5 in the previous section. The K and S
curves for these pigments can then be used to derive the spectral curves for
the mixtures using Equation 1. Finally, the colors of these mixtures can be
displayed on the computer monitor by converting the spectral curve to CIE
XYZ coordinates
and then transforming
into the RGB primaries of the
monitor. This ability to calculate correct pigment mixtures and display those
results accurately on the monitor means that we can now bring the world of
realistic color selection and mixture into computer graphics. For instance,
artists could use their experience with paints and palettes to select and mix
colors on the computer just as they would in the studio; they are no longer
limited to the RGB additive color selection methods that monitor technology
has imposed.
The difference between organic and inorganic paint mixtures provides a
good example of why it is critical to use Kubelka-Munk
theory to predict
pigment mixtures. This organic/inorganic
distinction is an important one for
artists; mixtures of white with inorganic paints tend to “gray” out much more
than those with organic pigments. For example, mixtures of napthol red (an
organic paint) with titanium dioxide and cadmium red (an inorganic paint)
with titanium dioxide can be compared. Although the red colors look very
similar in their masstone states, their respective mixtures with white are
quite different. This difference can be seen in Figure 7 where the effect is
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C. S. Haase and G. W. Meyer
Fig. 7. Canvas painted with real pigments showing mixtures of cadmium red (top)
and napthol red (bottom) with titanium white. From left to right the tint concentrations were 2, 5, 10, 20, 40, 80, and 100 percent by dry weight of pigment.
demonstrated using real pigments. These differences are known to those who
use paints; artists choose their palettes very carefully based on the types of
colors they would like to produce from paint mixtures.
The existing algorithms that are used in computer graphics to simulate
paint mixing are unable to capture this difference between organic and
inorganic pigments. Commercially available electronic paint programs use an
RGB instead of a spectral representation for the paints that are mixed. A
weighted average based upon the convolution mask of the brush profile is
taken between the RGB of the paint being applied and the RGB of the
background. Oddy and Willis 1221 have extended this approach to include a
pigment medium, but it is still an RGB averaging technique. For efficiency,
paint programs often use the non-gamma corrected
RGB values, which
means that the mixing cannot even be described as additive.
Given the RGB values for the pure cadmium red, napthol red, and titanium white pigments used in Figure 7, a pigment mixing simulation that
applies this weighted average RGB approach leads to the result shown in
Figure 8. Because there is no way to directly relate the pigment concentrations used to create Figure 8 to the proportions employed in the weighted
average
RGB approach, the color scales in this figure were computed by
linearly interpolating between the RGB values for titanium white and the
RGB values for each of the two pigments. The white pigment RGB to red
pigment RGB proportions are identical for corresponding swatches in the
two color scales, and the initial proportions were selected so that the beginACM Transactions on Graphics, Vol. 11, No. 4, October 1992.
Fragmented Materials for Realistic Image Synthesis
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317
Fig. 8. Result of using paint mixing approach employed by traditional painting programs to
simulate the mixture of cadmium red and napthol red pigment with titanium white.
ning colors in the real and the simulated cadmium red scales were close (an
exact match was not possible). Note that because both pigments in their
masstone states have almost identical RGB values, there is little difference
between the cadmium red and the napthol red color scales in the simulation.
Converting from an RGB to a CiWY representation has no effect on this
result because the transformation is linear. A wavelength-based approach
that models the pigments as cyan, yellow, and magenta filters is also incapable of capturing the mixing differences between the two pigments.
In order to correctly model pigment mixing in computer graphics, a
pigment-based approach, such as the Kubelka-Munk method, must be employed. This is the only way to include the absorption and scattering phenomena that take place in real paint films. An important step in this approach is
to obtain data for the pigments which is suited to the purposes of the
application. In our case, we were comparing our research results with artists’
oil paints, so we needed to get data for the pigments in a medium of oil.
Off-the-shelf oil paints can contain various ingredients in addition to the
main pigment, so we used pure dry pigment and linseed oil.
To obtain K and S data for the cadmium red and napthol red pigments, it
was first necessary to create masstone solutions for cadmium red, napthol
red, and titanium white; this was done by mixing linseed oil with the pure
pigments. Next, a tint with titanium white was created for each red pigment
ACM Transactions on Graphics, Vol. 11. No. 4. October 1992.
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C. S. Haase and G. W. Meyer
Nap Tit New Chart
1.00
Napthol Red Masstone
—
0.00
400
450
500
550
600
.—
650
700
Wavelength
Fig. 9. Spectralreflectancecurvesfor titaniumwhite,naptholred, and a tint of titaniumwhite
with naptholred.
by combining measured amounts of dry pigment with oil. Spectral measurements were then taken of all masstones and tints (Figure 9). From the
spectral curve for titanium white, we deduced K and S values for that
pigment by setting S equal to one at all wavelengths and solving for K. Using
these absorption and scattering values and the spectral curves for the masstones and tints for each red pigment, the K and S curves were calculated as
described in the previous section (Figure 10). Finally, given the data for all
three pigments, the spectral curves for arbitrary concentrations were calculated using Equation 1.
The results of the Kubelka-Munk
modeled canvas are seen in Figure 11.
This example compares favorably with the actual paint mixtures in Figure 7.
It is interesting to contrast this result with that shown for additive mixing in
Figure 8. With both the real mixtures and the Kubelka-Munk
modeled
mixtures, there is a marked difference between the two stripes; cadmium red
goes to white much faster than napthol red. But in the additive mixing
example, both pigments proceed at about the same rate and the difference at
any given concentration is not very significant. These differences are quantifiable, using a color difference formula. In Figure 12, we have plotted A E as
measured in CIE L*a*b* coordinates for the Kubelka-Munk
and additive
pictures. We can see that the color differences between the pigment scales in
the Kubelka-hfunk
experiment are much higher for the tints at smaller red
pigment concentrations.
But with the additive experiment, the color diflerACM Transactions on Graphics, Vol. 11, No. 4, October 1992.
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napthol ks new chari copy
Fig. 10.
K and S curves corresponding to napthol red pigment shown in Fig. 9
Fig. 11. Result of using Kubelka-Munk theory to simulate the mixture of cadmium red and
napthol red pigment with titanium white.
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12
Kubelka-Munk
10
Mixing
8
:6
4
2
J_____
1
2
RGB Mixing
4
3
Swatch
5
6
7
Number
Fig. 12. Color difference in CIE L*a*b * space between the cadmium
in Figure 8 and between the same color scales in Fig. 11.
red and napthol red scales
ences remain relatively flat, showing that the mixing differences for the
pigments were not picked up by the additive mixing approach.
Another experiment
that was performed for paint mixing calculations
involved modeling a color wheel which was painted using standard off-theshelf paints and then modeled using data for the pigments in those paints.
The color wheel (Figure 13) was thus constructed: the outer circle uses just
the three primaries-alizarin
red, prussian blue, and cadmium yellow
medium—as
marked on the canvas. These primaries were mixed in 50/50
proportions to derive secondaries, which lie halfway between each of the
primaries. Each secondary was then mixed with each primary to derive the
tertiaries, which complete the circle. The inner circle is composed of 50/50
mixtures of the primaries and secondaries with titanium dioxide, shown in
the middle of the wheel.
The possible gamut in CIE L*a*b* space that can be produced using these
four pigments was computed using Kubelka-Munk
theory and is shown in
Figure 14. For comparison, consider Figure 15 which shows the position in
CIE L*a*b* space of the color television monitor gamut from Figure 1.
Clearly the pigments are not capable of producing colors as saturated as
those created by the monitor. It is also interesting to note that a very dark
color is also not possible using the four pigments selected. A simulation of the
canvas shown in Figure 13 was also done and is included as part of Figure 19
which is discussed in Section 6. It should be noted that both the gamut in
Figure 14 and the canvas in Figure 19 were not computed using data that we
measured ourselves but from masstone and tint curves published elsewhere
[11]. As described in the previous section, such data must be used with care,
because of differences that may exist in materials and measurement techniques.
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Fig. 13. Canvas painted with real pigments showing mixtures of cadmium yellow (12
o’clock), alizarin red (4 o’clock), prussian blue (8 o’clock), and titanium white (center).
There are many potential uses for this approach to color mixing. One of
these applications is in computer painting programs; as the organic/inorganic mixing experiment pointed out, much more realistic results can be
derived by using these pigment mixing techniques than by using current
painting program mixing techniques. One such application that has been
developed is an airbrush program that uses the Kubelka-Munk approach to
color mixing to predict realistic mixing of colors on the painting canvas.
Instead of using arbitrary RGB colors for the painting tool, the user selects
colors from among a palette of available pigments (Figure 16). This palette is
limited only by the pigment data available to the program. When the user
selects a color, the K and S data for that pigment is loaded into the airbrush.
Painting then occurs in the usual paint program manner except that, where
colors overlap on the canvas, their mixture is correctly computed and displayed. This is accomplished by having each pixel, as well as the brush, keep
track of its own current K and S values. When any pixel is painted over, new
K and S values are calculated using the data for the brush and the pixel and
Equations 3 and 4. Then the reflectance of the pigment mixture is determined, the RGB tristimulus values are found, and the appropriate color is
displayed in that pixel.
This painting program uses a K and S frame buffer technique which stores
the current K and S values for each pixel at four specific wavelengths. These
wavelengths were selected to compute tristimulus values from spectral energy distributions by using Gaussian quadrature [19]. Limiting the wavelength information is necessary for the purposes of real time interaction and
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C. S. Haase and G. W. Meyer
Fig. 14. Gamut produced in L*u*b* space by mixing
varying concentrations of cadmium yellow, prussian blue,
alizarin red, and titanium white.
Fig. 15.
Color television monitor gamut in L*a*b* space.
conservation of memory resources. Since the pigment reflectances are quite
smooth, the errors caused by this low sampling rate are minimized. The
program also manipulates the floating point K and S values in appropriate
integer form to save memory resources. Future hardware developments may
make it possible to avoid these optimization methods and still maintain good
performance.
5. COLOR MATCHING
Most scene generation in computer graphics involves specification of object
color by RGB triplets; this method is certainly an easy way to pick a color,
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Fig. 16. Airbrush program showing pigment selection, current brush color,
and artwork being worked on.
Fig. 17. Use of a hue, saturation, and brightness color model to select a color to
be matched. Program mixes pigment concentrations to generate a reflectance that
will create the desired color.
and it is also readily interpreted by the computer during scene computation.
However, when color fidelity is critical and when rendering techniques, such
as ray tracing and radiosity are being used, a wavelength based approach to
specifying color is necessary. A database of spectral reflectances can be
created [20] to facilitate the use of such curves in image synthesis. Colors
available when such an approach is used are limited to the number of curves
in the database. The number of possible colors can be increased, and the
number of curves in the database can be decreased, by storing pigment data
instead of individual reflectances. Pigment mixtures can be determined by
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using the techniques described in the previous section. However, speci&ing
color in terms of pigment concentrations is not very intuitive for most people;
if a particular shade of red is imagined for some plastic object in the scene,
the typical computer user would probably not be able to come up with
appropriate pigment concentrations which would produce this particular red.
What is required is a method for interpreting the user’s intuitive color
designation and then producing a pigment specification.
This problem is
known in the pigment industry as “color matching.” In standard color matching problems, a sample of color is presented and a match is computed which
most closely approximates
that color. For instance, if you want to repaint
part of your house but do not have the original paint, the paint store could
give you a mixture which approximates
that color using potentially very
different pigments than were used originally. What we need to do in computer graphics matching is analogous. For example, in creating a graphics
image, the user may want to specify the color of a wall using RGB values
(which could be textually or graphically entered, depending on what user
interface tools were available for the scene description). Then the matching
program would convert this color into concentrations
of whatever pigments
provided the best match.
From the preceding sections of this article it is clear that, given a set of
pigments about which the proper data is known, we can predict the outcome
of mixtures of those pigments. Therefore it is reasonable to assume that we
can use these capabilities to calculate color matches with the pigments-that
is, given certain data about an arbitrary color, it is possible to specify which
pigments and concentrations
are required to approximate the closest match
possible to that color.
For this project, we wanted a program that would allow the user to specify
a color in terms of its color space coordinates, either RGB or CIE XYZ. An
important additional feature was to give the user a graphical interface for
this specification, so that the user did not have to know the actual coordinates but could choose the color interactively
and visually. The program
would also allow the user to specify various parameters and constraints for
the program to use in calculating the match, such as how close the match
should be and which set of pigments to use. Given this color specification, the
program would compute the best color match with the given set of pigments.
There are several possible approaches to this matching problem. They
differ in the types of data required for the computations, the computational
complexity involved, and the suitability of the results. The most straightforward approach to solving the problem is to iterate through the set of
available pigments, calculate their mixtures in all possible combinations and
concentrations,
and choose the mixture that comes the closest to matching
the sample. Although this brute-force method is certainly the slowest possible, it is also the most accurate; since all possible mixtures are checked, the
result is the most correct. In addition, this method provides the greatest
flexibility in terms of the data required for the calculations, because data
about the sample can be provided in any suitable form. The calculations
depend on the data only to the extent that it is needed to see how close a
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mixture has come to matching the sample. The search time can be limited by
various means, including: restricting the set of pigments being searched,
halting calculations when a mixture comes within some specified tolerance,
and having the program intelligently direct the search based on some criteria
such as pigment suitability. Although these methods can shorten the search
time significantly, a much more efllcient method is required.
Another approach to color matching is spectrophotometric
matching [2, 5,
17, 23, 27], in which the calculations attempt to find a match with a spectral
curve that best fits the spectral curve of the given color. While the algorithm
used for this approach is much more eflicient than the brute-force approach
described above, the data required for the calculations are not appropriate for
our application. Since the color we would like to match may be specified only
in terms of CIE XYZ or RGB, we do not necessarily have a characteristic
spectral curve to match against. Clearly, another method is needed.
Tristim UIUSmatching, a better approach, is the one taken in most of the
color matching references used in this research, including [2, 3, 13, 17, 18].
Just as in spectrophotometric
matching, this method intelligently
narrows
down the search using a steepest descent method of optimization to find the
best match for a set of pigments. But this method also has the advantage of
requiring that only the tristimulus coordinates of the sample color be matched.
Tristimulus matching makes a linear approximation
of the nonlinear functions involved in the Kubleka-Munk
equations, so it may require several
iterations to arrive at an answer (if a match is possible). Each iteration
results in a new trial match which is then compared to the sample using
some color difference formula. If the trial is not within a pre-specified
tolerance, then a new match is calculated. If at any time the concentration
values of any of the pigments become negative, then a match with these
pigments is not possible.
While this method is adequate for many cases of color matching, it has
some disadvantages.
Most significantly,
this approach as described in the
literature is limited to a given set of three pigments plus white. To use the
formulas as is, a general match from a larger set of pigments could only be
derived by iterating through the set of pigments, coming up with the best
match from all sets of three. But in a general color-matching
case, this may
be terribly inefficient; a large pigment database would require a substantial
calculation time to find the best match. Furthermore,
the user may not
necessarily want to be limited to three pigments in the resulting color match.
If a better match could be derived from a larger set of pigments, then it would
be helpful to provide that match instead.
Another problem with the tristimulus method is that the amount of change
required in the pigment concentrations
for the next match is based on the
absolute differences between the CIE XYZ coordinates of the current trial
match and the sample match; a color difference formula is used to judge
whether the trial is a good match, but CIE XYZ coordinates are used to
derive each new trial. The color difference formula would be a better metric to
use in deriving the trial matches as is done in Cogno et al. [7]. Finally, it is
important to account for metamerism in any color match. Metamerism is the
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degree to which a color changes in different lighting and viewing conditions.
Usually, we do not want a color to appear very different across environments,
so we would like to limit the degree of metamerism to some tolerance level.
A matching algorithm must therefore take into account an arbitrarily large
set of pigments, make better use of the color difference formula, and have the
ability to control the degree of metamerism. Its main goal, of course, is to find
a match which is as close to the sample color as possible. In other words, the
color difference should be equal to zero. Therefore, we need to minimize the
color difference function.
The CIE L*a*b* color difference formula [28] is:
AE = [( Aa*)2 + (Ab*)2
+ (AL*) 2] ’/2
(8)
where
()
L* = 116 ;
1/3
– 16
(9)
( 10)
a*=500[Ei3-Er31
b*=200[(~r3-(:r31
(11)
and
x, Y, z
tristimulus
values of the color in question
XO, YO, ZO
tristimulus
values of the light source
Aa*, Ab*, AL*
differences
in values between
colorants.
First we note that minimizing this value can also be achieved by minimizing
its square, A E2, thus we can eliminate the square root from Equation 8. To
find the minimum for A E2, any of several numerical methods may be used.
We used the method of steepest descent, which follows the negative of the
gradient at any given point on the function to get as close to zero as possible.
This is an iterative method, like the tristimulus method explained above,
which takes guesses at minimum values and then refines these guesses by
repeated application of the same equation. Both this method and the tristimUIUSmethod depend on some initial guess. Allen [1] describes an algorithm to
derive an intelligent first guess based on spectrophotometric
data. Lacking
this type of data, we use an arbitrary guess for the first trial, as suggested by
Billmeyer and Saltzman [6].
An important constraint which complicates the situation is that the concentrations of the pigments should remain positive. The tristimulus method, as
described in the references, does not attempt to find the closest match.
Instead, if a match within the specified tolerance cannot be found, its
calculations return negative values for some pigment concentrations, indicating that a match is not possible. Our method, however, should be able to
ACM Transactions on Graphics, Vol. 11, No. 4, October 1992.
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obtain the closest match possible. This ability is especially important for
what we are trying to accomplish—matching
RGB colors with their closest
subtractive approximations;
many colors in RGB space may not have exact
matches with any combination
of pigments. Instead of using the positive
concentration value as a constraint on the problem, though, we can rewrite
the concentration value c, as the square of some value q,. This will allow the
equations to be solved for the optimal qi values while the Ct values remain
positive. Using this technique allows us to keep the values of the concentrations positive without adding a complicating constraint to the problem.
Now we can proceed with finding the gradient for A E2. This will be the
direction vector toward the minimum. For the gradient we will need the
partial derivatives of the formula with respect to the q, of each of the
pigments involved. We know that:
~AE~
I?AE2
—
_—
x 2q*.
(?Ct
~q,
(12)
Next, by the Chain Rule:
(?AEP
~AE2
~c
= ——+
dx
1
These component
(7X
dAE2 dY
——+——
dY
(?C,
ac,
(?AE2
az
dZ
dc, “
(13)
partials are written as follows:
(14)
/IAE2
dY
–lOOO Aa* + 232 AL* + 400Ab*
=
( 15)
3Y2’’3X;’3
dAE2
—
dz
.
–400Ab*
(16)
3z2/3zy3
and
(17)
( 18)
(19)
where
(?R
~=
1
SM K, – KM S,
H
s;
x 1l-&
’20)
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and
k
loo/’ZAsAyA
‘A,
observer
yA, 2A
SA
matching
functions
given light source
R,
spectral curve of match
KM, SM
K and S for the current match trial
K,, Si
K and S for the ith pigment
A
wavelength; our method uses 16 wavelengths,
increments from 400 to 700 nm.
spaced at 20 nm
Given the current match trial, the gradient is just a vector containing the
values for the left side of Equation 12. Some multiple of this gradient is
added to each current q, which are then used to derive new values for each
Ci. These new Ci values are used to calculate a new match trial. If the match
is within a specified tolerance of the sample or if a match is not possible but
the nearest minimum has been reached, then the program is finished. If not,
the process is repeated.
The factor by which the gradient is multiplied has a severe effect on the
calculation process, determining how fast an answer can be reached and how
good that answer is. For our program, we based the multiplier on a scalar
factor of the current color difference between the sample and the match;
trials that were further away would take longer steps than those closer in.
This method helps the program avoid overshooting the minimum and having
to backtrack. The other important consideration for the multiplier is for those
cases in which no perfect match is possible; in these cases, we would like to
derive the closest minimum. The gradient multiplier helps this process. If the
program ever overshoots the minimum, it is clear that the current step size is
too large. So a scaling factor is input into the equation which decreases the
gradient multiplier by half. Overshooting again causes another halving of the
scalar multiplier. In this way, the closer the program gets to a minimum,
whether it is a match or not, the smaller steps it will take. Execution then
continues until either a match or a minimum has been found.
A further important enhancement can be made to this approach to account
for metamerism.
In order to keep the degree of metamerism
as low as
possible, the function to be minimized could be rewritten as some combination of equations of the form of Equation 8, where each equation was
calculated for a different light source or viewing condition. Then minimizing
this new formula would result in a match which was minimized across these
different conditions. The effects of metamerism
upon the result could be
easily changed by weighting the different components of the equation appropriately. For example, let us suppose that we wished to achieve a match
under CIE standard illuminant C but also wished to minimize metamerism
for CIE standard illuminant A. Let us further suppose that the metamerism
component of the match was only ~ as important as the initial match. Then
our minimizing equation would be A E2 = 4 A E$ + A E;.
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The approach we took toward color matching provided two major advantages over the standard color matching techniques covered in the references.
First, we minimize in terms of a color difference formula, so the results we
get are the closest color match in terms of this formula. This is in contrast to
the tristimulus approach, which minimizes over X, Y, and Z and thus only
achieves a minimum in tristimulus space, which does not necessarily correspond to a minimum color difference. Second, our formula admits an arbitrary number of pigments and attempts to find a suitable match from that
set, regardless of how many pigments are to be used. This is helpful in many
cases where either many pigments are required to make a match or the user
does not know which limited set of pigments would be adequate. The tristimUIUSmethod formula, on the other hand, requires there to be a set number of
pigments. In the usual form, this number is only three, though more could be
added for different light sources [13, 17]; in this case, the user would have to
know ahead of time which pigments were suitable or the machine would have
to iterate through a larger set of pigments until an appropriate match were
found.
There are some things about this approach, however, that would require
further development for a full-featured color matching algorithm. First, this
steepest descent approach is guaranteed to find a minimum, but it will not
necessarily be the global minimum; i.e., if there are many pigments involved
and the initial guess is far from the true global minimum, the equations
might settle on a local minimum even though a better match might exist with
some other combination.
This problem can be easily addressed through
pigment choice and through the initial guess, although a more automated
way of achieving this is desirable. Also, the equation does not take into
account the cost of the pigments, which is important in many color matching
applications.
Finally, there is currently no way to limit the number of
pigments used in the resulting match, other than the total number of
pigments involved; if the user inputs 100 pigments into the program, then the
result could potentially be a mixture which uses a small concentration from
each pigment.
We developed a program using the above techniques and specifications.
The user inputs several pieces of information for the search to begin. To
choose the color to be matched, the user can input color coordinates either
textually using CIE XYZ or RGB coordinates or graphically (Figure 17).
Next, the user may choose which pigments should be used in determining the
match. The tolerance level, which is the color difference in CIE L*a* b* space
below which the computer considers the match successful, can also be specified by the user. All of these selections are handled via a dialog box (Figure
17).
Using this information,
the program uses the method described above,
iterating through several trial matches until a minimum is found. Data for
the closest match is displayed on the screen, including concentration values
for each of the chosen pigments, the color difference, the resulting CIE XYZ
values, a color swatch of the match color, and windows displaying the
spectral curve and color of each trial used in deriving the match (Figure 18).
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Trial
1
]
Trial
8
G. W. Meyer
/
———
400
450
500
—-—.
550
-—
600
650
700
Wavelength
Fig. 18. Final result ofasuccessful
match bythecolor
matching pro@amfor
acolorselectidin
the manner shown in F]g. 17. The pigments used for this match were titanium white, prussian
blue, cadmium yellow and alizann red. Successive attempts by the program to create the match
are shown.
6. CONCLUSION
The purpose of this work was to demonstrate
the capabilities of realistic
pigment modeling in the field of computer graphics. Techniques used in the
pigment industry have been modified to work within the domain of computer
graphics to facilitate more realietic computer color specification. The opening
sections of this paper motivated and validated the approach taken here. The
mixing techniques discussed in Section 4 provide the basis for better, more
realistic painting programs and other tools which require mixing and displaying pigmented materials. Finally, the matching techniques
discussed in
Section 5 show the feasibility of combining intuitive color selection with
realistic color computation in generating realistic computer images.
As a final example, consider the image in Figure 19. Using the matching
and mixing programs described in this article we can derive spectral curves
for arbitrary pigment mixtures. These curves can then be used to specify
realistic colors for materials in computer generated scenes. Every color in the
scene depicted in Figure 19 was described by a characteristic spectral curve.
Most of the objects, except the RGB monitor picture and the white canvas,
use curves which represent real pigment mixtures calculated by our programs. For example, the color wheel on the canvas is based upon the
pigments used to create the artwork shown in Figure 13. With the exception
of the color television picture, notice that unrealistic highly saturated colors
do not occur. The range of colors possible has been constrained in a realistic
way by the selection of a limited number of pigments and by the accurate
simulation of their mixing properties.
ACM Transactions on Graphics, Vol. 11, No. 4, October 1992.
Fragmented Materials for Realistic Image Synthesis
Fig. 19. Image demonstrating the use of pigment mixing and color matching
techniques. All surfaces had spectral retlectances determined from pigment mixtures.
APPENDIX
The Kubelka-Munk equations for complete hiding given in Section 3 were
used to perform the pigment mixing work described in this paper. They are
derived here from the initial assumptions because this detail is not available
in most of the widely available references.
The situation as it occurs in a given pigmented solution is shown in Figure
20. In this figure, the pigmented solution is a paint, applied in a uniform
thickness of x over a substrate of reflectance R,. Let i be the incident light
which is transmitted down through the paint, either directly from the source
or by scattering of light off of pigment particles. Let j be the light which is
returning to the surface, either by being reflected off of the substrate or
scattered from pigment particles.
Suppose we are examining some very thin sub-layer of the paint, of
thickness dx. The amount of light that is lost from i through the layer is:
(K + S)icLx
(21)
and the amount lost from j is:
(K + S)jdx
where S is the scattering component and K is the absorption. Now we must
add back to i and j that light which was scattered from light coming in the
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\
A
dx
i
j
\‘
x
4
T
Substrate
of Reflectance
Fig. 20. Coordinate system used to calculate
inside a pigmented surface.
f3g
light energy scattering
and absorption
other direction; i.e., the light that was scattered from the incident light is
added to the light returning to the surface and vice versa. Then we have:
di=(K+S)idx–
–dj=
(K+
S)jdx
Sjdx
(23)
–Sidx.
(24)
Note that since i and j represent light vectors traveling in opposite directions, the change of their signs through the film are opposite,
The solution to Equations 23 and 24 begins by setting a = (1 + K/S) and
reexpressing them as
di
—=~i–j
S dx
(25)
– dj
Sdx
Adding these two equations
together
—
i2Sdx
and rearranging
leads to
–
–2a$
+ ‘~
+ 1.
(27)
“2
+ ‘~ – + 1.
(28)
1
Rule, we get:
d(j/i)
—
S dx
Setting
(26)
.2
idj –jdi
From the Quotient
=aj–i.
.
‘
–2a<
L
r = j/i then gives us
dr
Sdx
‘r2
–2ar+l
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and therefore
dr
S J dx.
/ r’ –2ar+l=
(30)
We would like to know the value of the change in r as the thickness varies
from O to some arbitrary thickness t.At a thickness of O, the value of r will
simply be the reflectance of the substrate, R~. At a thickness of t, the
reflectance is equal to some R. Using integration by partial fractions we have
1
dr
R
f fi~rz –2czr+l=
~
dr
1
dr
R
–J
+b)–2b~~r–(a–
–/
2b~~r–(a
b)
(R-a
-b)(R~-a+b)
— ~ in
2b
(R–a+b)(R~–a–b)
where b = ~az – 1. Therefore,
from Equation
(R-a
-b)(R~-a+b)
A In
2b
(R–a+b)(R~–a–b)
(31)
30
= St.
(32)
The particular thickness that we are interested in solving for is t = x, as
this thickness provides complete hiding of the substrate. We can solve for the
reflectance at this thickness, R,, as t - X:
~n(R-a-b)(R~-a+b)
(R-a
(R-a
For the following
+b)(R~
-b)(R~-a+b)
~2.sth
-a-b)
=( R-a+
= 2Stb
b)(RX
(33)
-a-b).
(34)
substitutions,
R=R,
RK=O
we get:
1
R. =
a+fi
—
‘+%-T”
(35)
This form is one variation of the Kubelka-Munk
formula for computing R,.
The more common forms can be obtained from this by rearrangement and are
given in the text as Equations 1 and 2,
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ACKNOWLEDGMENTS
The authors would like to thank Robert Gamblin of Gamblin Oils for his help
in understanding
the world of artists’ paints, Michael Ken y for writing the
Airbrush pigment painting program shown in Figure 16, Christy Wiseman
for creating the artwork displayed in Figure 16, Linda Peting for developing
the color space visualization program that generated Figures 1, 4, 3, 5, and
14, Sohail Mohammad for modeling the scene in Slides 8 and 11, and David
Collett for his mathematical
assistance in obtaining the color matching
method used in Section 5. The images shown in Slides 8, 11, and 19 were
produced using the Radiance software package developed by Greg Ward at
the Lawrence Berkeley Laboratories and modified by Roy Ramberg at the
University of Oregon to handle light on a wavelength-by-wavelength
basis.
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